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Abstract

We generalize a theorem of Nymann that the density of points in ZdZd that are visible from the origin is 1/ζ(d)1/ζ(d), where ζ(a)ζ(a) is the Riemann zeta function View the MathML source∑i=1∞1/ia. A subset S⊂ZdS⊂Zd is called primitive if it is a ZZ-basis for the lattice Zd∩spanR(S)Zd∩spanR(S), or, equivalently, if S can be completed to a ZZ-basis of ZdZd. We prove that if m points in ZdZd are chosen uniformly and independently at random from a large box, then as the size of the box goes to infinity, the probability that the points form a primitive set approaches 1/(ζ(d)ζ(d−1)⋯ζ(d−m+1))1/(ζ(d)ζ(d−1)⋯ζ(d−m+1)).